M. Heydari-Malayeri - Paris Observatory

1) The fact that the Sun, which contains 99.9% of the mass of the
→ solar system, accounts for about 2% of the total
→ angular momentum of the solar system. The problem of outward
→ angular momentum transfer has been a main topic of interest for
models attempting to explain the origin of the solar system.
2) More generally, in star formation studies, the question of the origin of the angular momentum
of a star and the evolution of its distribution during the early
history of a star. Consider a filamentary molecular cloud with a length of 10 pc and a
radius of 0.2 pc, rotating about its long axis with a typical
→ angular velocity of Ω = 10-15 s-1.
At a matter density of 20 cm-3,
the cloud is about 1 → solar mass.
The cloud collapses to form a star with
radius of 6 x 1010 cm. The conservation of angular momentum
(∝ ΩR2) requires that as the radius decreases from 0.2 pc to
the stellar value, a factor of 107, the value of Ω must increase by 14
orders of magnitude to 10-1 s-1. The star's
rotational velocity will be 20% the speed of light and the ratio of
→ centrifugal force to gravity at the equator will be about
104. Observational data, however, indicate that the youngest
stars are in fact rotating quite slowly, with rotational velocities of
10% of the → break-up velocity. The angular momentum problem
was first studied in the context of single stars forming in isolation (L. Mestel,
1965, Quart. J. R. Astron. Soc. 6, 161). For more information see,
e.g., P. Bodenheimer, 1995, ARAA 33, 199; H. Zinnecker, 2004, RevMexAA 22, 77;
R. B. Larson, 2010, Rep. Prog. Phys. 73, 014901, and references therein.

The impressive discrepancy of about 120 orders of magnitude between the theoretical
value of the → cosmological constant and its observed value.
→ Quantum field theory
interprets the cosmological constant as the density of the
→ vacuum energy. This density can be derived from
the maximum energy at which the theory is valid, i.e.
the → Planck energy scale (1018 GeV).
The theoretical vacuum → energy density is
(1018 GeV)4 = (1027 eV)4 = 10112
erg cm-3. On the other hand, the observed vacuum energy density is estimated
to be about (10-3 eV)4 = 10-8
erg cm-3. There is, therefore, a discrepancy of about 120 orders of magnitude.

A computer experiment that was aimed to study the
→ thermalization process of a
→ solid.
In other words, the goal was to see whether there is an
approximate → equipartition of energy
in the system, which would mean that the motion is
→ chaotic.
Using computer simulation, Fermi-Pasta-Ulam studied the behavior of
a chain of 64 mass particles connected by → nonlinear
springs.
In fact, they were looking for a theoretical physics problem
suitable for an investigation with one of the very first computers,
the he MANIAC (Mathematical Analyzer, Numerical Integrator and
Computer). They decided to study how a → crystal
evolves toward → thermal equilibrium
by simulating a chain of particles,
linked by a quadratic interaction potential, but also by a weak
nonlinear interaction.
Fermi-Pasta-Ulam assumed that if the interaction in the chain
were nonlinear,
then an exchange of energy among the normal modes would occur, and
this would bring forth the equipartition of energy, i.e. the
thermalization.
Contrary to expectations, the energy revealed no tendency toward
equipartition. The system had a simple quasi-periodic behavior,
and no → chaoticity
was observed. This result, known as the Fermi-Pasta-Ulam paradox,
shows that → nonlinearity
is not enough to guarantee the equipartition of energy
(see, e.g., Dauxois et al., 2005, Eur. J. Phys., 26, S3).

The observed fact that the → geometry of the
→ Universe is very nearly flat, in other
words its density is very close to the → critical density.
This would be an extreme coincidence because a → flat Universe
is a special case. Many attempts have been made to
explain the flatness problem, and modern theories now include the idea
of → inflation.

A problem with the standard cosmological model of the Big Bang related
to the observational fact that regions of the Universe that are
separated by vast distances nevertheless have nearly identical
properties such as temperature. This contradicts the fact that light
moves with a finite speed and, as a result, certain events which occur
in the Universe are completely independent of each other. Inflationary
cosmology offers a possible solution.

1) Given the trajectory of a particle moving in a → central force
field, determine the
law governing the central force.
2) Inversely, considering a central force -k/r2, determine the
trajectory a particle moving in the field will take.

Low-mass → protostars are about an order of magnitude less luminous
than expected. Two possible solutions are that → low-mass stars
form slowly, and/or protostellar → accretion
is episodic. The latter accounts for less than half the missing luminosity. The solution
to this problem relates directly to the fundamental question of the time required to form
a low-mass star (McKee & Offner, 2010, astro-ph/1010.4307).

A problem concerning the compatibility of grand unified theories
(→ GUTs) with standard
cosmology. If standard cosmology was combined with grand unified theories,
far too many → magnetic monopoles
would have been produced in the early Universe. The
→ inflation hypothesis aims at
explaining the observed scarcity of monopoles. The inflation has
deceased their density by a huge factor.

The mathematical problem of solving the equations of motions of any number of
bodies which interact gravitationally. More specifically, to find their
positions and velocities at any point in the future or the past, given their
present positions, masses, and velocities.

The observed underabundance, by one or two orders of magnitude, of
→ dwarf galaxies orbiting
→ spiral galaxies compared to their number predicted by the
standard model. The → cold dark matter (CDM) model
predicts that dwarf galaxies are the building blocks of large galaxies like the Milky Way
and should largely outnumber them. Dwarf galaxies form first,
they merge into bigger and bigger galaxies, and galaxies into groups of galaxies.
The dark matter halos, however, are very dense, and dwarf halos are not destroyed in
the merging, resulting in their large predicted number, in numerical simulations.

The mathematical problem of studying the behavior (e.g., velocities, positions)
of any number of objects moving under their mutual gravitational attraction for
any time in the past or future. Same as the → many-body problem.

A major discrepancy between the flux of neutrinos detected at Earth from the solar core
and that predicted by current models of solar nuclear
fusion and our understanding of neutrinos themselves.
The problem, lasting from the mid-1960s to about 2002, was a considerably lesser
detected number of neutrons compared with theoretical predictions.
The discrepancy has since been resolved by new understanding of
neutrino physics, requiring a modification of the
→ standard model of particle physics,
in particular → neutrino oscillation.

The mathematical problem of studying the positions and velocities of
three mutually attracting bodies (such as the Sun, Earth and Moon) and the stability
of their motion. This problem is surprisingly difficult to solve, even in the
simple case, called → restricted three-body problem,
where one of the masses is taken to be negligibly small so that
the problem simplifies to finding the behavior of the mass-less body in the
combined gravitational field of the other two. See also
→ two-body problem, → n-body problem.

In classical mechanics, the study concerned with the dynamics of
an isolated system of two particles subject only to the Newtonian gravitational
force between them. The problem can be separated into two single-particle problems
with the following solutions. The equation of the
→ center of mass is
governed by the equation of the same form as that for a single particle.
Moreover, the motion of either particle, with respect to
the other as origin, is the same as the motion with
respect to a fixed origin of a single particle of → reduced mass
acted on by the same internal force. See also
→ three-body problem,
→ n-body problem.